We need to look at our schooling system if we really want to fix this problem.
Unfortunately education is completely immersed in the social fabric of our culture and is very much tied-up with how Brits define themselves (part of our class system). This means that taking a dispassionate look at our education system is pretty much impossible for any political institution or individual as there is always the accusation of unfairly favouring this or that socio-economic group. This of course is not to mention the fact that traditional 'academic' success is seen in Britain as the only real type of education, most people's views of vocational education are, in fact, quite shockingly backward. However, I don't see any of this changing in the near future in England (at least, if not the UK as a whole) so we will still have this problem for a while yet.
That said, if I were to have a new education system then it would probably look a bit like the Swiss/German systems with selected education with multiple entry points and good funding across the board (not just for the 'academic' institutions) so that we can create a high quality skilled labour force as well as university graduates. I would cut defense budgets to fund this new education system.
I would also require much more stringent tests for non-state funded schools to be able to claim charity status, specifically to allow VAT exemption on fees. People should have the choice to send their children to private institutions, but they should accept that this is a luxury (or indeed value-added activity) and should be taxed as such unless that institution can demonstrate that the majority of its work is of a truly charitable and beneficial nature, not just that it is effectively a not-for-profit venture. This would have the impact of either increasing tax revenues or encouraging the private education sector to do more to benefit wider society (I hope it would do the latter as I feel that would be more beneficial).